y is equal to five minus three x over x squared plus three x

And we want to figure out

what's the derivative of y in respect to x.

Now it might immediately jump out at you that look,

y is being defined as a rational expression here

as the quotient of two different expressions.

We could even view this as two different functions.

You could view this one up here as u of x,

so you could say this is the same thing,

this is the same thing as u of x, over,

you could view the one in the denominator as v of x.

That one right there is v of x

and so if you're taking the derivative of something

that can be expressed in this way,

as the quotient of two different functions,

well then you could use the quotient rule.

I'll give you my little aside, like I always do

the quotient rule, if you ever forget it

it can be derived from the product rule

and we have videos there,

cause the product rule's a bit easier to remember.

But what I can do is just say,

"look, dy dx, if y is just u of x over v of x",

I'm just gonna restate the quotient rule.

This is going to be, this is going to be

the derivative of the function in the numerator, so.

d, dx of u of x

times the function in the denominator

times v of x minus, I'll do the

minus the function in the numerator, u of x,

times the derivative of the function in the denominator

times d dx, v of x

and we're almost there

and then over, over the function in the denominator squared

the function in the denominator squared.

So this might look messy but all we have to do now

is think about what is the derivative of u of x?

What is the derivative of v of x?

And we should just be able to substitute those things

back into this expression we just wrote down.

So let's do that.

So the derivative with respect to x of u of x, of u of x

is equal to, let's see, five minus three x.

The derivative of five is zero.

The derivative of negative three x,

well that's just gonna be negative three.

That's just negative three.

If any of that look completely unfamiliar to you

I encourage you to review the derivative properties

and maybe the power rule.

Now let's think about what is the derivative

with respect to x, derivative with respect to x,

of v of x, of v of x?

Well derivative of x squared,

we just bring that exponent out front,

it's gonna be two times x to the two minus one

or two x to the first power or just two x

and then the derivative of three x is just three.

So two x plus three.

Now we know everything we need to substitute back in here.

The derivative of u with respect to x,

this right over here is just negative three.

V of x, this we know is x squared plus three x.

We know that this right over here is v of x.

And then u of x we know is five minus three x.

Five minus three x.

The derivative of v with respect to x

we know is two x plus three, two x plus three.

And then finally v of x, we know is x squared plus three x.

So this is x squared plus three x

and so what do we get?

Well we are going to get, it's gonna look a little bit hairy

It's going to be equal to negative

I'll focus this so first we have this business up here.

Negative three times x squared plus three x.

So I'm just gonna distribute the negative three.

So it's negative three x squared minus nine x

and then from that we are going to subtract

the product of these two expressions and so let's see,

what is that going to be?

Well, we have a five times two x,

which is ten x

a five times three which is 15

we have a negative three x times two x

so that is going to be negative six x squared

minus six x squared, and then a negative three x times three

so negative nine x.

And let's see we can simplify that a little bit.

Ten x minus nine x,

well that's just going to leave us with x.

So ten x minus nine x is just going to be x

and then in our denominator, we're almost there.

In our denominator we could just write that as

x plus three x, x squared plus three x squared

or if we want we could expand it out,

I'll just leave it like that

x squared plus three x squared

and so if we wanna simplify

or attempt to simplify this a little bit.

It's going to be negative three x squared minus nine x

and then you're gonna have a negative minus x

minus x and then minus 15

and then minus negative six x squared

so plus six x squared, all of that over

x squared plus three x squared

or x squared plus three x, squared.

I should say it that way.

Now let's see this numerator I can simplify a little bit.

Negative three x squared plus six x squared

that's going to be positive three x squared

and then we have in orange,

we have negative nine x minus an x,

well that's gonna be minus ten x, minus ten x

and then we have minus 15.

So minus 15.

So there you have it, we finally have finished.

This is all going to be equal to

this is all going to be equal to

three x squared minus ten x minus 15

over x squared plus three x, squared

and we are done.